First Principles AI

(i)

Solve:

(x+iy)^2=25+60i

\begin{aligned} (x+iy)^2 &= x^2-y^2+2xy\,i \\ &=25+60i \end{aligned}

So:

\begin{aligned} & x^2-y^2=25 \quad (1)\\ & 2xy=60 \Rightarrow xy=30 \quad (2) \end{aligned}

Also,

\begin{aligned} (x^2+y^2)^2 &= (x^2-y^2)^2+(2xy)^2 \\ &=25^2+60^2=4225 \\ \Rightarrow\ x^2+y^2&=65 \quad (3) \end{aligned}

Add/subtract (1) and (3):

\begin{aligned} 2x^2&=90 \Rightarrow x^2=45 \Rightarrow x=\pm 3\sqrt{5}\\ 2y^2&=40 \Rightarrow y^2=20 \Rightarrow y=\pm 2\sqrt{5} \end{aligned}

Since $xy=30>0$ , $x$ and $y$ have the same sign.

\boxed{(x,y)=(3\sqrt{5},\ 2\sqrt{5})\ \text{or}\ (-3\sqrt{5},\ -2\sqrt{5})}